Open Problems in Integrable Systems — Around the Birkhoff conjecture

Definitions and mathematical statement

In the usual planar Euclidean billiard interpretation, let $\Omega\subset{\mathbb{R}}^2$ be a bounded domain with a sufficiently smooth boundary, and let $B_\Omega$ be its billiard map on non-grazing collision states: a state specifies a boundary point and an inward direction, and the next state is obtained by straight travel and specular reflection. The supplied extract does not explicitly state this dimension, strict convexity, or boundary-regularity class; these must be checked in the source context.

A genuine two-periodic orbit consists of distinct states $z_0,z_1$ with \[ B_\Omega(z_0)=z_1,\qquad B_\Omega(z_1)=z_0. \] The return map near one state is $P=B_\Omega^2$, which fixes $z_0$. A natural precise reading of the question asks whether one can choose $\Omega$, this orbit, an angle $\alpha\in{\mathbb{R}}/(2\pi{\mathbb{Z}})$, and a local conjugacy $H$ sending $z_0$ to $0$, so that as germs \[ H\circ P\circ H^{-1}=R_\alpha,\qquad R_\alpha(u,v)=(u\cos\alpha-v\sin\alpha,\ u\sin\alpha+v\cos\alpha). \] This interpretation uses rotation of a planar neighborhood and the second iterate. The literal source instead says the billiard map is conjugated near the two-periodic orbit to rigid rotation; it does not specify the iterate or whether a conjugacy is taken around one state or around the two-state orbit as a whole. That distinction is left explicit rather than silently identifying $B_\Omega$ with its return map.

The conjugacy category is also missing: a local homeomorphism, a smooth diffeomorphism, an analytic diffeomorphism, and a symplectic conjugacy give different problems. The allowed angle is not restricted to be irrational or nonresonant, and no exclusion of $\alpha=0$ or other rational multiples of $2\pi$ is given. The question asks for existence for some angle, not for every prescribed angle. Exact local conjugacy is the requested issue; a finite-order or formal normal form alone would not specify that conclusion.

The supplied summary labels the question open or unverified and relates it to local billiard normal forms, twist maps, and Birkhoff/KAM questions, but supplies no result matching this exact statement. This is an attributed source-verification limitation, not an independently established literature status.

Short English statement

Can a billiard table be chosen whose local dynamics around a two-bounce orbit are exactly a rigid rotation after a coordinate change? The likely return-map formulation is written explicitly, while the source's missing iterate, conjugacy regularity, domain class, and angle restrictions remain unresolved. No exact linearization result is supplied by the record.

Sources

• Local source registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $2100404$, source code catalogue entry, including the supplied OPEN-TRIAGE summary. Dataset attribution: Ulam AI and UnsolvedMath Contributors; original-source attribution and rights are retained.
• Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018), Problem 4.4: https://arxiv.org/abs/1804.03737. The metadata reports source-TeX extraction and July 29, 2026 access, with current-status and rights review flags; no author list is supplied in this record.
• The supplied summary mentions Birkhoff normal forms, twist maps, KAM-type rigidity, and the Birkhoff conjecture literature, but provides no precise additional citation or verified answer to this exact local-conjugacy question. No linked source was inspected for this draft.